Electromagnetic (EM) structures can often be characterized, at least approximately, by a linear response function that relates input signals to output signals. For example, a microwave or radio-frequency (RF) structure (i.e., an electromagnetic device designed for operation at signal wavelengths in the microwave or radio-wave regime) may be represented as a black box that is excited by transmission lines, and described in terms of a scattering matrix that relates ingoing transmission line modes to outgoing transmission line modes. Electromagnetic structures excited by transmission lines are hereinafter referred to as signal integrity (SI) structures. Generally, each transmission line of an SI structure includes two or more distinct conductor traces, one of which is usually considered to be a local reference ground, whereas the other ones are considered to be signal traces. In a black-box representation, the interfaces of the SI structure with the transmission lines are termed ports, and the intersections of the individual signal traces with the ports are called terminals. FIG. 1A illustrates an exemplary SI structure 100 that includes two planar ports 102, 104, each containing three terminals, and FIG. 1B schematically depicts a black-box representation of that structure.
The response of an SI structure to an input signal generally depends on the geometry, materials, and operating frequency of the structure (hereinafter referred to as “design variables”). Simulating an SI structure across a range of values of the design variables in a manner that facilitates comparisons between the resulting responses represents a difficult problem.
In addition to changing the design variables of the SI structure to be analyzed, engineers often want to model the SI structure as embedded in a network or fixture, or—conversely—computationally remove the effect of such a network or fixture. This is called “de-embedding.” Another important capability of a good simulation is “renormalization,” which accounts for mismatches between the material properties or geometries of the external transmission lines and those of the internal transmission lines. These requirements further complicate the already challenging problem of modeling a highly complex system to permit engineers to vary system designs and, without actually building those designs, predict and understand their real-world behavior.